# Scientific Notation Calculator

This scientific notation calculator converts between scientific notation, e-notation, engineering notation, and real numbers. To use the calculator, please provide a number in scientific notation, e-notation, engineering notation, or real number format then click the "Calculate" button.

## What is scientific notation?

Scientific notation is a method for writing numbers that is intended to make it easier to communicate very large or very small numbers. For example, the number 0.0000000000002 has 12 zeros. This number is very difficult to read quickly, as is the number 5819237232. As a result, it can be easy to make mistakes in determining the magnitude of numbers written this way, particularly when many of the digits are 0. Since the use of numbers is widespread in scientific contexts, scientific notation was developed to help combat the inefficiency and potential inaccuracy of communicating numbers purely in decimal form.

Scientific notation expresses numbers as a product of a real number in decimal form multiplied by a power of 10:

m×10^{n}

where m is a non-zero real number, and n is the power to which 10 is raised. There are many different ways that m can be selected, but most typically, it is selected such that the absolute value of m is at least 1, but less than 10. For example, for the number 5,127, we would select m = 5.127. To determine n, count the number of digits before or after the decimal point; for numbers less than 1, n is the number of digits before the decimal point, and we would add a negative sign to indicate that the number is smaller than 1; for numbers larger than 1, n is the number of digits after the decimal point. Thus, for this example, we would count the number of digits after the decimal point to find that there are 3 digits, and 5,127 can therefore be written in scientific notation as:

5.127×10^{3}

Although this is the convention, we could also have written 5,127 in scientific notation as any of the following:

51.27×10^{2}

512.7×10^{1}

The process of converting to scientific notation is the same for numbers less than 1. Using our previous example, 0.0000000000002 can be written in scientific notation as:

0.0000000000002 = 2×10^{-13}

Notice that the exponent very quickly tells us the magnitude of a number, while writing the number in decimal form would require us to count the digits to determine how small the number is. Similarly, 5819237232 in scientific notation is:

5819237232 = 5.819237232×10^{9}

The 10^{9} term very quickly tells us that this number is in the billions range without requiring us to count the number of digits.

## What is scientific e-notation?

Scientific e-notation is very similar to scientific notation. The key difference between the two is instead of writing the 10^{n} term, we simply use e to represent the "×10^" portion. Thus, 5.127×10^{3} is written in scientific e-notation as:

5.127e3

Note that sometimes "E" is used instead of "e," but they both represent the "×10^" portion of the expression. In general form, a non-zero real number, m, and the power to which it is raised, n, is written in scientific e-notation as:

men = m×10^{n}

Selecting m and n in scientific e-notation uses the same method as described in the "What is scientific notation" section.

## What is engineering notation?

Engineering notation is another version of scientific notation. It uses the same format as scientific notation,

m×10^{n}

with the exception that m and n must be selected such that n is divisible by 3. Engineering notation makes it possible to relate powers of 10 to their SI prefixes (discussed in more detail below). For example, the number 45000000 in scientific notation is:

4.5×10^{7}

In engineering notation, we would instead write this as:

45×10^{6}

The SI prefix that corresponds to 10^{6} is mega, so given that the number is in units of meters, the above can be read as "forty-five megameters." In contrast, we would read the number in scientific notation as "four point five times ten to the seven meters." Engineering notation therefore facilitates reading and oral communication of numbers.

### SI prefixes

SI prefixes are the prefixes used in the International System of Units (SI) to denote a multiple or submultiple of an SI unit. For example, the "kilo-" in "kilometer" indicates 10^{3} meters, while the "nano-" in "nanometer" indicates 10^{-9} meters. The table below shows the SI prefixes divisible by 3 and their corresponding powers of 10, listed from smallest to largest.

Prefix | Symbol | Power |
---|---|---|

yocto | y | 10^{-24} |

zepto | z | 10^{-21} |

atto | a | 10^{-18} |

femto | f | 10^{-15} |

pico | p | 10^{-12} |

nano | n | 10^{-9} |

micro | μ | 10^{-6} |

milli | m | 10^{-3} |

kilo | K | 10^{3} |

mega | M | 10^{6} |

giga | G | 10^{9} |

tera | T | 10^{12} |

peta | P | 10^{15} |

exa | E | 10^{18} |

zetta | Z | 10^{21} |

yotta | Y | 10^{24} |